C. Mae Waugh Barrios
Final Course Reflection Assignment
December 19, 2014
If you had asked me before June 2014 about chips, I would have replied, “Potato or tortilla?” Likewise, I thought a number line was something you did while square dancing. Just kidding. But I really had no idea of their significance in integer instruction and conceptual knowledge.
Before this course, all I knew about integers were the rules. I knew a negative times a negative equals a positive. I knew a negative plus a negative equals a negative. I knew to subtract integers, you just add the opposite. In fact, when I was in seventh grade, I remember our integer assignment was to write a narrative story with integers as the characters, in order to demonstrate that we knew the rules. I also remember getting a poor grade on the assignment, because my sixth-grade brain couldn’t keep the rules straight, because they really held no meaning for me.
I had heard about the chip model before our summer workshop, and even knew that yellow meant positive and red meant negative, but I had never used it before and I had never seen it in action. Completing operations using the chips and chip mats was completely new to me, but struggling through the process this summer was an enriching experience. What really struck me the most was how chips are able to help represent why the integer rules exist. This helped me make meaning of the rules I had half-wittedly memorized so many years ago.
I had used number lines before, but only to build foundation for integer value and relationship, to show that negatives were less than zero. I’d never used it for any operations. Actually the first time I saw the vector model for addition was when I was co-teaching a math class and we were doing practice MCAS questions. With my numeracy knowledge as an adult (and math teacher) I was able to figure out what the diagram meant, but I had never been explicitly taught the meaning behind it. As we went over it at the October workshop, I felt conflicted over the procedure of always starting the vectors at zero. I thought it was an additional step that complicated the process and might only confuse my students. But when I actually taught the vector model to my students, I found it was an essential step for students to differentiate between addition and subtraction models on the number line.
However, the activity that resonated most with me over the entire workshop was the integer clothesline. While it doesn’t specifically relate to integer operations, having the numeracy understanding of the placement and relationship among integers and rational numbers is an integral part to building conceptual understanding for students. I used the integer clothesline for all of my middle school math ELD students and it was impactful in so many ways. First, it was kinesthetic and got even my most active students engaged and involved. Second, it was a form of pre-assessment, so that I was able to determine my students’ background knowledge and experience with integers, because they come from schools from all over the world, with different instructional styles and pacing. And finally, it was an activity that provided immediate feedback to the students, not only me but also my students’ peers. If a student began putting an integer in the wrong place, his classmates were the ones to help him revise his placement.
These three integer models helped me connect the rules and concepts I had learned in isolation to a foundation of conceptual knowledge and they gave me resources so that I could help my students do the same.
Obviously in order to successfully compute integer operations, students need number sense and operational fluency. They need to know how to add, subtract, multiply and divide multi-digit whole and partial numbers and they need to know what these operations actually mean. As I began teaching my students about multiplying integers, I asked them what 3x4 “meant” and all my students answered, “12.” They had memorized math facts, which is definitely important for computational fluency, but they were missing the conceptual foundations. If they didn’t know 3x4 meant 3 groups of 4, how could they model 3x4 using the chips? How could they model it using a number line? Additionally, using models helps students build the conceptual knowledge that subtraction is really addition of negative numbers. I told the students subtraction actually doesn’t exist and they couldn’t believe it! I challenged them to give me a subtraction problem that I couldn’t change into an addition problem, and of course they were unable to stump me.
Having an understanding of integers and integer operations is an important foundation as the students move toward algebra and then on to calculus.
Yet none of the models were a perfect method solitarily and the students struggled with difficulties and misconceptions with each one. All models have their limitations and their benefits. It was difficult for the students to transition from adding using the vector model to subtracting using the vector model. While the vector model helped to denote the directionality of each operation and integer, manipulating the vectors from one operation to the next was so confusing for the students. They asked me why the process was so different and it was hard to get them to understand that subtraction is really finding the distance between two integers on the number line and addition was combining their values. Without this differentiation, the sums and differences were neither accurate nor understood.
The Walking the Number Line activity proved to help the students to model operations with vectors on the number line. The Walking the Number Line activity has the students assume the role of integer and they move in the negative and positive direction, adding and subtracting positive and negative numbers. It helps the student build the understanding of the directionality of the negative and positive numbers. These skills then transferred to students’ paper number line representations.
Additionally, I learned to make sure students understand one operation model completely, before moving on to the next operation. That way students feel competent and confident with addition, for example, before moving on to subtraction, so they are less likely to confuse the two models.
Using models to build conceptual knowledge helps the students to build meaning for the reason why the mathematical rules work. Rules are there for convenience, not to take the place of actual understanding and knowledge.